#7621047

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1260 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,23,28)(19,24,31,39)(20,25,33,41)(21,26,30,38)(22,27,35,43)(29,37,45,47)(32,40,36,44)(34,42,46,48);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,31)(18,24,28,39)(20,29,33,45)(21,35,30,22)(25,37,41,47)(26,43,38,27)(32,34,36,46)(40,42,44,48);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,23,33)(18,25,28,41)(19,29,31,45)(21,36,30,32)(22,34,35,46)(24,37,39,47)(26,44,38,40)(27,42,43,48);(10,13)(14,16)(17,21,22,19,23,30,35,31)(18,26,27,24,28,38,43,39)(20,32,34,45,33,36,46,29)(25,40,42,47,41,44,48,37);(1,4,5,3)(2,7,6,8)(9,11,12,15)(10,14,13,16)(20,33)(25,41)(29,45)(32,36)(34,46)(37,47)(40,44)(42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=gh,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7528243);